If a Banach function lattice \(X\)  has the Fatou property and \(X_a = X_b\) , then the Köthe dual \(X^{\prime }\)  is isometrically isomorphic to \((X_a)^*\)  [1]}. Moreover, there exists a unique decomposition of a bounded linear functional \(F = H + S\) , where \(H\)  and \(S\)  are regular and singular linear functionals, respectively [2]}. Since any singular functional returns zero for every order-continuous element, we see that \(X^* = (X_a)^* \oplus (X_a)^{\perp }\)  is isometrically isormorphic to \(X^{\prime }\oplus X_{s}^*\) , where \(X_{s}^*\)  denotes the set of all singular functionals on \(X\) . Hence, we can say that the order-continuous subspace \(X_a\)  is an M-ideal in \(X\)  if \(\Vert F\Vert _{X^*} = \Vert H\Vert _{X^*} + \Vert S\Vert _{X^*} = \Vert h\Vert _{X^{\prime }} + \Vert S\Vert _{X^*}\) .
